Letf(x)=(-1).CMRB=f(n).
x
(-1)
1/x
x
∞
∑
n=1
MKB=t.
N∞
2N
∫
1
πt
1/t
t
ThenproveRe(MKB)=Im+1t=Im-1t.
∞
∫
0
f(1-t)-f(1+t)
2πt
∞
∫
0
f(1+t)+f(1-t)
2πt
NoMKB=Im-1t.
∞
∫
0
f(1-t)-f(1+t)
2πt
ReMKB+NoMKB-CMRB=0.
Andletg(x_)=.
1/x
x
ThenproveIm(MKB)=-t.
N∞
2N
∫
0
g(1-t)+g(1+t)
2
πt
Show any relationship between NoMKB and Im(MKB) to relate CMRB with MKB.
MKB=t.
N∞
2N
∫
1
πt
1/t
t
MKB=t
N∞
2N
∫
1
πt
1/t
t
In[]:=
f[x_]=(-1)^x(x^(1/x)-1);
CMRB=N[NSum[f[n],{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision100],50]
Out[]=
0.18785964246206712024851793405427323005590309490014
ReMKB=Im[N[NIntegrate[(f[(1-It)]-f[(1+It)])/(Exp[2Pit]+1),{t,0,Infinity},WorkingPrecision100],50]]
Out[]=
0.070776039311528803539528021830282001365754696203363
ReMKB-Im[N[NIntegrate[(f[(1-It)]+f[(1+It)])/(Exp[2Pit]-1),{t,0,Infinity},WorkingPrecision100],50]]
Out[]=
0.×
-51
10
NoMKB=Im[N[NIntegrate[(f[(1-It)]-f[(1+It)])/(Exp[2Pit]-1),{t,0,Infinity},WorkingPrecision100],50]]
Out[]=
0.117083603150538316708989912223991228690148398696776
ReMKB+NoMKB-CMRB
Out[]=
0.×
-51
10
In[]:=
g[x_]=x^(1/x);
MKB=N[NIntegrate[Exp[IPit]g[t],{t,1,1Infinity},WorkingPrecision100],50]-I/Pi
Out[]=
0.07077603931152880353952802183028200136575469620336-0.68400038943793212918274445999266112671099148265500
ImMKB=-(N[INIntegrate[(g[(1-tI)]+g[(1+tI)])/(2Exp[Pit]),{t,0,Infinity},WorkingPrecision100],50]+I/Pi)
Out[]=
0.×-0.68400038943793212918274445999266112671099148265500
-67
10
ReMKB=N[INIntegrate[(g[(1-tI)]-g[(1+tI)])/(2Exp[Pit]),{t,0,Infinity},WorkingPrecision100],50]
Out[]=
0.070776039311528803539528021830282001365754696203363
ImMKB+ReMKB-MKB
Out[]=
0.×+0.×
-51
10
-51
10